Mathematical Modeling of Waves in a Non-linear Shell with Wiscous Liquid Inside It, Taking into Account Its Movement Inertia

Abstract

New mathematical models of wave movements in infinitely long physically non-linear shell are constructed. The shell contains viscous incompressible liquid, considering its movement inertia. The models are based on the hydrodynamics related problems, described by shells and viscous incompressible liquid dynamics equations in the form of the generalized MKdV equations. The effective numerical algorithm with the use of the Grobner bases technology was proposed. It was aimed at constructing differential schemes to solve the generalised MKdV equation, obtained in the given article. The algorithm was used to analyze deformation non-linear waves propagation in elastic and non-elastic cylinder shells with viscous incompresible liquid inside them. The numerical experiments were carried out on the basis of the obtained numerical algorithm. The experiments made it possible to reveal new viscosity and incompressible liquid inertia effects on the deformation wave behavior in the shell, depending on Poisson ratio of the shell material. In particular, the exponential growth of wave amplitude under the liquid presence in the shell made of non-organic materials (various pipelines and technological constructions) is revealed. In the case of organic materials (blood vessels) viscous liquid impact leads to wave quick going out. Liquid movement inertia presence leads to deformation wave velocity transformation. Supported by grants RFBR 19-01-00014-a and the President of the Russian Federation MD-756.2018.8.